Travail Pratique 2: Risk Management

1 Loading the data

The file Market.rda contains the list “Market”.

The first element of the list is an xts object corresponding to the value of the SP500 index from “2000-01-03” to “2013-09-10”.

The second element (vix) corresponds to the value of the VIX index at the same dates.

The third element (rf) corresponds to the term structure on “2013-09-10” for 14 maturities varying from 1 day to 30 years. The term structure for the annualized interest rates can be seen in Figure 1.1 .

Figure 1.1: Interest rates term structure.

The fourth (calls) and the fifth (puts) elements of the list contain the strikes (K), maturities (tau) and implied volatilities (IV) for calls and puts options respectively.

We plot the interpolated volatility surface in Figure 1.2 .

Figure 1.2: Volatility surface for the options in the list Market.

2 Pricing of a portfolio of options

The prices of the call options using the Black-Scholes formula are given in table 2.1 .

Table 2.1: Prices of the (call) options in our book

S0	K	tau	sigma	r	C
1683.99	1600	0.08	0.1453	0.00115	87.569
1683.99	1650	0.08	0.1453	0.00115	47.724
1683.99	1750	0.16	0.1453	0.00144	15.302
1683.99	1800	0.16	0.1453	0.00144	6.380

The price of this book (the sum the call prices that constitute the book) is \(V_0\) = 156.98.

3 Procedure for the P&L computation

We start by simulating the 5 days ahead log-returns of the SP500 and the VIX (if necessary). By aggregating them we get the 5 days compounded log-returns that will be used to simulate the corresponding underlying values of the SP500 and the VIX.

We then use our Black & Scholes option pricing function to compute the call prices and ultimately evaluate the book-values of our 10000 simulated scenarios.

The P&L is equal to the difference between the discounted 5 days ahead book-values \(BV_{5d}\) and our initial book-value namely \(V_0\).

4 One risk driver and Gaussian model

In this section, we assume that the risk of our book of options is driven by a single factor namely the change in price of the SP500 index. The log-returns of the index are assumed to follow a normal distribution with parameters given by the mean and the standard deviation from the historical values.

Figure 4.1: P&L distribution for normal log-returns of the SP500. The red line at -122.77 is the value at risk at 95% probability.

We simulate 10000 values for the stocks in 5 days and use them to estimate the distribution for the P&L. The distribution of the P&L is given in Figure 4.1 .

The VaR at risk at the 95% confidence level is VaR95 = -122.77 and the expected shortfall is ES95 = -134.95. So we expect the book of options to lose at least 122.77 from its initial value 5% of the time.

5 Two risk drivers and Gaussian model

In this section, we assume that the risk of our book of options is driven by two factors namely the change in price of the SP500 index and the volatility of the market (given by the VIX). The log-returns of the SP500 and the log-returns of the VIX are assumed to follow a bivariate normal distribution with parameters given by the mean and the variance-covariance matrix computed from the historical values.

Figure 5.1: P&L distribution when log-returns of SP500 and VIX are assumed to follow a bivariate normal distribution. The red line at -111.59 represents the value at risk at 95% probability.

We simulate 10000 values for the SP500 and the VIX in 5 days and use them to estimate the distribution for the P&L. The distribution of the P&L is given in Figure 5.1 .

The VaR at risk at the 95% confidence level is VaR95 = -111.59 and the expected shortfall is ES95 = -123.7. So we expect the book of options to lose at least 111.59 from its initial value 5% of the time.

6 Two risk drivers and copula-marginal model (Student-t and Gaussian copula)

In this section, we assume that the risk of our book of options is driven by two factors namely the change in price of the SP500 index and the volatility of the market (given by the VIX). The log-returns of the SP500 are assumed to follow a Student-t distribution with \(\nu = 10\) degrees of freedom and the log-returns of the VIX are assumed to follow a Student-t distribution with \(\nu = 5\) degrees of freedom. A normal Copula is assumed to link the marginals.

Figure 6.1: P&L distribution when log-returns of the SP500 and the VIX are assumed to have Student-t marginals and their marginals to be linked by a Gaussian copula . The red line at -99.02 represents the value at risk at 95% probability.

We simulate 10000 values for the SP500 and the VIX in 5 days and use them to estimate the distribution for the P&L. The distribution of the P&L is given in Figure 6.1 .

The VaR at risk at the 95% confidence level is VaR95 = -99.02 and the expected shortfall is ES95 = -113.06. So we expect the book of options to lose at least 99.02 from its initial value 5% of the time.

7 Volatility surface

Figure 7.1: P&L distribution when log-returns of the SP500 and the VIX are assumed to follow a bivariate normal distribution. The simulated values of the VIX are shifted by the one year ATM volatility difference. The red line at -108.6 represents the value at risk at 95% probability.

We simulate 10000 values for the SP500 and the VIX in 5 days and use them to estimate the distribution for the P&L. The simulated values of the VIX are then shifted by the difference between the one year ATM value of the implied volatility and today value of the VIX.

The implied volatility model is : \(\sigma(m, \tau) = \alpha_1 + \alpha_2(m - 1)^2 + \alpha_3(m - 1)^3 + \alpha_4\sqrt{\tau}\)

The distribution of the P&L is given in Figure 7.1 .

The VaR at risk at the 95% confidence level is VaR95 = -108.6 and the expected shortfall is ES95 = -121.15. So we expect the book of options to lose at least 108.6 from its initial value 5% of the time.

8 Full approach

Figure 8.1: P&L distribution when log-returns of the SP500 follow a GARCH(1,1) and the log-returns of the VIX an AR(1). The invariants of the two processes are linked by a Gaussian copula. The red line at -71.41 represents the value at risk at 95% probability.

Here, we fit a GARCH(1,1) to the log-returns of the SP500 and an AR(1) process to the log-returns of the VIX. The residuals of the two processes are used as invariants and are linked by a Gaussian copula. We generate 10000 values for the SP500 and the VIX in 5 days and use them to estimate the distribution for the P&L. The distribution of the P&L is given in Figure 8.1 .

The VaR at risk at the 95% confidence level is VaR95 = -71.41 and the expected shortfall is ES95 = -84.55. So we expect the book of options to lose at least 71.41 from its initial value 5% of the time.